November  2016, 15(6): 2203-2219. doi: 10.3934/cpaa.2016034

Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

1. 

Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal

Received  December 2015 Revised  July 2016 Published  September 2016

The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.
Citation: Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034
References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, Contemporary Math., 526 (2010), 125-150. doi: 10.1090/conm/526/10379.  Google Scholar

[2]

M. Beals, Self-spreading of singularities for solutions to semilinear wave equations, Annals Math., 118 (1983), 187-214. doi: 10.2307/2006959.  Google Scholar

[3]

M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge, Comm. Pure Appl. Anal., 13 (2014), 1669-1683. doi: 10.3934/cpaa.2014.13.1669.  Google Scholar

[4]

S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities, Int. Math. Res. Notices, 5 (1996), 201-220. doi: 10.1155/S1073792896000153.  Google Scholar

[5]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[6]

M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations, J. AMS, 17 (2004), 297-359. doi: 10.1090/S0894-0347-03-00445-4.  Google Scholar

[7]

V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in (2+1)-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296. doi: 10.1063/1.524669.  Google Scholar

[8]

H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, Adv. Diff. Equ., 19 (2014), 359-386.  Google Scholar

[9]

H. Pecher, Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge, Adv. Diff. Equ., 20 (2015), 1009-1032.  Google Scholar

[10]

S. Selberg, Multilinear Space-time Estimates and Applications to Local Exisatence Theory for Nonlinear Wave Equations, PhD. Thesis Princeton, 1999.  Google Scholar

[11]

S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation, Int. Math. Res. Not., 2008, art. ID rnn107.  Google Scholar

[12]

S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[13]

T. Tao, Multilinear weighted convolutions of $L^2$-functions and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 838-908.  Google Scholar

[14]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Diff. Equ., 189 (2003), 366-382. doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[15]

J. Yuan, Global solutions of two coupled Maxwell systems in the temporal gauge, Discr. Cont. Dyn. Syst., 36 (2016), 1709-1719. doi: 10.3934/dcds.2016.36.1709.  Google Scholar

show all references

References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, Contemporary Math., 526 (2010), 125-150. doi: 10.1090/conm/526/10379.  Google Scholar

[2]

M. Beals, Self-spreading of singularities for solutions to semilinear wave equations, Annals Math., 118 (1983), 187-214. doi: 10.2307/2006959.  Google Scholar

[3]

M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge, Comm. Pure Appl. Anal., 13 (2014), 1669-1683. doi: 10.3934/cpaa.2014.13.1669.  Google Scholar

[4]

S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities, Int. Math. Res. Notices, 5 (1996), 201-220. doi: 10.1155/S1073792896000153.  Google Scholar

[5]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[6]

M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations, J. AMS, 17 (2004), 297-359. doi: 10.1090/S0894-0347-03-00445-4.  Google Scholar

[7]

V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in (2+1)-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296. doi: 10.1063/1.524669.  Google Scholar

[8]

H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, Adv. Diff. Equ., 19 (2014), 359-386.  Google Scholar

[9]

H. Pecher, Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge, Adv. Diff. Equ., 20 (2015), 1009-1032.  Google Scholar

[10]

S. Selberg, Multilinear Space-time Estimates and Applications to Local Exisatence Theory for Nonlinear Wave Equations, PhD. Thesis Princeton, 1999.  Google Scholar

[11]

S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation, Int. Math. Res. Not., 2008, art. ID rnn107.  Google Scholar

[12]

S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[13]

T. Tao, Multilinear weighted convolutions of $L^2$-functions and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 838-908.  Google Scholar

[14]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Diff. Equ., 189 (2003), 366-382. doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[15]

J. Yuan, Global solutions of two coupled Maxwell systems in the temporal gauge, Discr. Cont. Dyn. Syst., 36 (2016), 1709-1719. doi: 10.3934/dcds.2016.36.1709.  Google Scholar

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